Thermomagnetic properties of DyAl 2 single crystal, calculated by ATOMIC MATTERS MFA Computation System

Transcription

1 Theroagetc propertes of DyAl sgle crystal, calculated by ATOMIC MATTERS MFA Coputato Syste RAFAŁ MICHALSKI, JAKU ZYGADŁO INDUFORCE M.Pszoa 1/9 31- Cracow POLAND JAGIELLONIAN UNIVERSITY, Faculty of Matheatcs ad Coputer Scece, Łojasewcza Cracow, POLAND Abstract: We preset the results of calculatos of propertes of DyAl sgle crystals perfored wth our ew coputato syste called ATOMIC MATTERS MFA [1,]. A localzed electro approach ethodology was appled to descrbe the electroc structure evoluto of Dy os over a wde teperature rage ad estate Magetocalorc Effect (MCE). Theroagetc propertes of DyAl were calculated based o the fe electroc structure of the f 9 cofgurato of the Dy ato. Our calculatos yelded: agetc oet value ad drecto; sgle-crystalle agetzato curves zero feld ad exteral agetc feld appled varous drectos (T, ext ); the f-electroc copoets of specfc heat c f (T, ext ); ad teperature depedece of the agetc etropy ad sotheral etropy chage wth exteral agetc feld - S(T, ext ). The cubc uversal CEF paraeters values used for all CEF calculatos was take fro [3] ad recalculated for uversal cubc paraeters set for the RAl seres A +7.1Ka 0 ad A Ka 0. Magetc propertes were foud to be asotropc due to cubc Laves phase C15 crystal syetry. These studes reveal the portace of ultpolar charge teractos whe descrbg theroagetc propertes of real f electroc systes ad the effectveess of a appled self-cosstet olecular feld calculatos for agetc phase trasto sulato. Key-Words: - DyAl, ReAl, Laves Phase, CEF, MFA, MCE, Atoc Matters. 1 Itroducto Codesed atter scece research advaced agetc refrgerat aterals has focused o the gat agetocalorc effect sce ts dscovery. Ths has advaced the developet of ear rooteperature agetc coolg techology. The RAl (Rrare earth eleet) c opouds are well kow teretallc aterals wth terestg agetc propertes at low teperatures. The agetc propertes of ths seres were studed tesvely the 1970s, but a ew wave of terest ths copoud faly has appeared due to research to aterals wth a large Magetocalorc Effect (MCE) for future agetc refrgerators. We preset the results of sulatos perfored by our ATOMIC MATTERS coputato syste [1,] of theroagetc propertes of DyAl. A few calculato results for the DyAl copoud are copared wth experetal data take fro the lterature [3-8]. All the lathades cobe wth aluu to for a RAl copoud wth the sae crystalle structure. Ths structure s the so-called cubc Laves phase C15 ad the pot syetry for the rare earth o s 3. Fg. 1. Cubc eleetary cell of RAl Laves phase C15 crystals. The structure was aed after the Swss crystallographer who perfored work o t he prototype copoud MgCu. The CEF paraeters descrbg the ultpolar charge teracto of R os the crystal surroudg ths structure were agreed for both copouds ad establshed by a E-ISSN: Volue 1, 017

2 prevous study [3,,9] for A +7.1Ka 0 ad A Ka 0. Coputato syste All DyAl calculatos were perfored wth ATOMIC MATTERS MFA coputato syste [], a exteso of the ATOMIC MATTERS applcato [1] that descrbes fe electroc structure ad predcts basc agetc ad spectral propertes of aterals a p araagetc state. ATOMIC MATTERS MFA calculates agetc, caloretrc ad spectroscopc propertes of atoclke localzed electro systes uder the fluece of Crystal Electrc Feld (CEF), sp-orbt couplg, ad agetc teractos, take both as Mea Feld Approxato (MFA) self-cosstet, dyac calculatos ad the fluece of establshed exteral agetc felds ext []. ATOMIC MATTERS MFA calculates acroscopc propertes of aterals defed teperature regos, especally aroud phase trasto teperature, such as: agetc oet (T, ext ) (sp ad orbtal, drectoal copoets); localzed electro specfc heat c f (T, ext ); localzed electro etropy wth a useful tool set for MCE; sotheral etropy chage S(T, ext ) calculatos; evoluto of eergy level postos; total free eergy. oth calculato systes cota advaced Graphc User Iterface (GUI) wth syste of herarchcal tabs for calculato results aageet, 3D teractve vsualzatos of potetals ad felds based o O pe GL graphc ege, tools ad databases for coveet ad effectve work. More -date forato about Atoc Matters coputato systes are avalable o our web page [10]. 3 Theoretcal backgroud The theoretcal approach appled at the heart of ATOMIC MATTERS MFA coputato syste s deeply rooted atoc physcs. Takg to cosderato the dvdual populato of states of fe electroc structure of os/atos at dfferet teperatures accordg to L. oltza statstcs ake t possble to defe the teperature depedeces of sgle oc propertes. ATOMIC MATTERS MFA ca sulate phase trastos of oc/atoc systes accordg to dyac calculatos of the olecular feld ol, sply defed as: ol(t) ol (T) (1) Ths teracts wth os to duce ther agetc oets. Such self-cosstet calculatos ca oly be perfored after establshg the olecular feld factor ol that s closely related to the teperature of phase trastos T C. For rapd calculatos a therodyacally stable teperature rego, ATOMIC MATTERS offers the followg CEF+Sp-Orbt+Zeea Haltoa accordg to the chose calculato space of groud ultplet J,Jz> or groud atoc ter LSL z S z > respectvely[1]: H H + H ˆ O (J, J ) + g J CEF Zeea z L J or H LS H CEF + H S-O + H Zeea ˆ O (L,L z) + λ L S + ( L + ge S) ext ext () (3) For all Haltoas: deotes CEF paraeters, O are Steves operators, λ-s the sp orbt costat, ad g L ad g e are the gyroagetc factors of a whole o or sgle electro respectvely. For a whole o or electro respectvely, s the ohr ageto ad ext s the exteral agetc feld. I all cases, calculatos the LSL z S z > space are ore physcally approprate due to ther copleteess, but tradtoal calculatos wth base JJ z > ca be also perfored by our coputato systes for coparsos ad rapd estatos []. For calculatg propertes teperatures aroud the agetc phase trasto pot, a self-cosstet ethodology for olecular feld calculato called Mea Feld Approxatos (MFA) s appled. The dea behd ths ethod s the estato of drecto ad value of the agetc feld (olecular feld) geerated by os at a defed teperature, ad calculato of the fluece of ths agetc feld for electroc state structures of os. I a selected calculato space, accordg to eq.1 we defe a olecular feld as a expected value of the total oet of the f electroc subshell ultpled by the olecular feld, ter oc exchage factor ol : ol ol g L < J > () or ol ol < L + g e S > (5) Where g L ad g e are the gyroagetc factors. O the bass of the calculated electroc level structure E (T), the drectoal copoets of agetc oets are establshed for all detcal os. Ths eas that Haltoa atrx dagoalzato s perfored for all defed teperature steps recurretly, cotrast to sple ATOMIC MATTERS calculatos [1], whch dagoalze atrces oe te for a sgle ru ad deduce all therodyac propertes fro the stable dscrete eergy level structure obtaed. Ths self- E-ISSN: Volue 1, 017

3 cosstet procedure provdes teperaturedepedet eergy level structure ad has oe oly free paraeter, ol, called the olecular feld paraeter. The value of ol s closely related to the phase trasto teperature T C of the acroscopc structure of os. The foral expresso of the full Haltoa used by ATOMIC MATTERS MFA coputato syste, accordg to the chose calculato space JJ z > or LSL z S z > respectvely, has the for: H H J ol or LS ol HJ + Hol + Oˆ (J, J ) ol gl J < J > + < J > + gl J Oˆ 1 + ol ( L + ges) < L + ges > + < L + ges > + ( L + g S) e (L,L ) + λl S + ext z z ext () + (7) The egevectors of the Haltoa are descrbed accordg to the selected calculato base by the total oetu quatu ubers JJ z > or sp ad orbt quatu ubers LSL z S z >. Usg the coutato relatos of the agular oetu operators, we obta forato about expected values of the projectos of agetc oetu of all electroc states at a chose teperature []: α gl E (T) ( T ) J exp (8) J Z(T) k T < > α α E (T) ( T ) L ges exp (9) LS Z(T) k T < + > α α Where: α dexes drectoal copoets, ubers the Haltoa egestates, whle Γ represets the expected value of the total agular oetu alog the α-axs the -th state: <J α> <Γ (T) J α Γ (T)> (10) <L α+g e S α > <Γ (T) L α+g e S α Γ (T)> (11) All property calculatos ca be doe for 3D (x,y,z) real space by usg coplex Haltoa atrx eleets defed by full expressos of exteded Steves O operators[11]. Mostly for coparso wth tradtoal calculato results, ATOMIC MATTERS also offers a D (x,z) calculato ethodology of a splfed odel of CEF teractos defed by Steves O operators wth real uber atrx eleets oly [1]. Takg to cosderato the possblty of the theral populato of states, we autoatcally obta the theral evoluto of sgle o propertes of the whole copoud [13,1]. Uder the therodyac prcple at teperature T0K oly the groud state s occuped. I ths stuato, the agetc oet of the o s exactly equal to the oetu of the groud state. If the teperature rses, the probablty of occupyg hgher states creases accordg to oltza statstcs. The uber of os wth eergy E wth a syste at teperature T s: E (T) exp k T N ( T ) N 0 (1) Z ( T ) where N ol -1 (Avogadro costat) ad Z(T) s the su of states. Kowg the su of the states, we ca detere the Helholtz free eergy F(T): F ( T ) k T l Z(T) (13) Accordg to therodyac prcples, the cotrbuto of localzed electros to the total specfc heat of aterals ca be calculated by uercal dervato of Helholtz free eergy: F ( T ) col( T ) T (1) T Ths akes t possble to calculate etropy accordg to the well-kow defto: S( T ) S( 0 ) + T 0 c(t) T dt (15) The value of electroc etropy for a defed teperature s dffcult to copare, but the sotheral chage of the etropy of the syste at a gve teperature s a very portat ateral paraeter that descrbes ts theroagetc propertes. Isotheral Etropy chage - S(T, ext ), captured for dfferet teperatures uder the fluece of dfferet agetc felds, s oe the ost portat propertes of a ateral that descrbes ts usefuless as a agetocalorc ateral. T he value - S(T, ext ), whch was extracted fro experetal specfc heat easureets, s ofte preseted as a basc descrpto of the Magetocalorc Effect (MCE) of a ateral [3-7]. I our approach, sotheral etropy chage ca be drectly calculated fro eq.15 accordg to the defto: S(T, ext ), S(T, ext 0)-S(T, ext ). (1) ATOMIC MATTERS MFA also provdes sgleoc agetocrystalle asotropy calculatos that clude full calculatos (wthout rllou fucto approxato) of K (T) agetocrystalle E-ISSN: Volue 1, 017

4 costats for defed teperature rages accordg to the relatos [15]: 3 0 ( ˆ 0 ˆ 0 ) ( ˆ 0 ˆ K1(T) O O 5 O 3 O ) 1 0 ( ˆ 0 ˆ O 5 O ), 35 0 K (T) ( ˆ 0 ˆ ˆ O O + O ) ( ˆ 0 0 ˆ 5 ˆ + O O + O ), 8 * 1 0 K (T) ( ˆ 0 5 ˆ O + O ), ˆ 0 * 11 K (T), K (T) ˆ 3 O 3 O. (17) 1 1 Where: Ô deotes theral expected values of Steves operators we defed accordg to C. Rudowcz [11]. The exchage teractos sulated accordg to MFA ethodology defed by eq. 1 pr ovde sulated propertes strogly depedet o oly oe paraeter, ol, whch s closely related to the teperature of phase trastos T C. It s easy to fd the value of ol for correct T C, but the value of ths paraeter ca be estated accordg to De Gees scalg [1]: T C ~G(f ), G(f )(g L -1) J(J+1) (18) De Gees scalg s also a useful tool for ol estato, as charge surroudgs ca be trasferred betwee os seres. The CEF part of Haltoa cotas Steves CEF paraeters. The values of these paraeters are oly approprate for the defed o. The recalculato of paraeters defed for a o A the crystal lattce surroudg of o the sae crystalle posto follows the sple schee: (o A) A (o ) (19) Steves paraeters ca be expressed by uversal A paraeters, accordg to the calculato space used, as follows: J,J z >; (J,J z )θ (J) <r f > A L,S,S z,l z >; (L,L z )θ (L) <r f > A (0) Where values of,, th power of average radus of f shell <r > have bee calculated by ay authors usg Hartree-Fock-type ethodology, ad the results ca be easly foud the lterature. The θ paraeters are the J or L depedet Clebsh- Gorda- type factors, soetes called α.β,γ Steves factors θ α, θ β, θ γ, whch ca be expressed by fte equatos avalable [17,18]. The values of <r f > are stored the syste s ope database together wth a referece. I all calculatos, we used the <r f > values tabulated [1]. The ablty to recalculate CEF paraeters betwee os ad calculato spaces offers a uque chace to establsh a acceptable splfcato of ethodology. The recalculato of CEF paraeters Atoc Matters systes s fully autoated, but a explct Steves Factors Calculator s also avalable. Calculato results The CEF paraeters used for DyAl were orgally estated by H.G. Purws ad Leso [] ad cted by A. L. La et al. [3]. The paraeters were provded cubc Steves otato: (5.5±1.) ev ad (5.±0.8) ev. We recalculated the values of Steves paraeters to obta uversal, o-depedet CEF otato A accordg to eq.19 ad eq.0. The paraeters A +7.1Ka 0 ad A Ka 0 obtaed ths aer defe charge surroudgs of R o a crystal lattce of RAl. A vsualzato of ths potetal s show fg.. T he tragular shapes that are coected to the locato of the coordatg Al os are located syetrcally cubc surroudgs ad reflect the ato posto the eleetary cell of Laves phase show fg.1. Fg.. Crystal Feld potetal vsualzato of cubc surroudg of R-os RAl defed by CEF paraeters: A +7.1Ka 0 ad A Ka 0. The vsualzato of postve sg potetal s cut off by the sphere. The teretallc copoud DyAl has strog agetocalorc propertes at low teperatures. Ths E-ISSN: Volue 1, 017

5 copoud has bee extesvely studed both theoretcally ad experetally [3-]. I ths secto, we preset the results of a vestgato of the agetc ad agetocalorc propertes of a DyAl sgle crystal. We have attrbuted the agets of DyAl to the Dy os ad calculated the fe electroc structure of the f 9 electroc syste cubc syetry takg to accout the crystal feld ad ter-ste, spdepedet exchage teractos. The eergy level schee derved s assocated wth the reducto of the degeeracy of the lowest atoc ter H gve by Hud s frst two rules. The full calculated eergy level structure the L,S,Lz,Sz> calculato space s show fg 3. The obtaed overall splttg s strogly depedet o the stregth of sp-orbt tra-atoc teractos. We use the free-o value of sp orbt costat of Ho 3+ os λ-550k ad obta a overall splttg of H atoc ter of about 15000K 1.3 ev. Detals of groud states structure are show fg.. Fg.. Groud ultplet eergy level structure vs. teperature calculated for f 9 electroc syste of Dy os DyAl. At Cure teperature T C 5K, the structure splts uder the fluece of olecular feld ad CEF. I the absece of a exteral agetc feld, the duced olecular feld splts ad oves to degeerated states. The value of the olecular feld factor establshed for DyAl, whch reproduces T C well at about 3K, s ol 3.T/. Above T C, ateral s a paraagetc state ad the groud state of Dy os s degeerated. The groud quartet cossts of two quas doublets. The wave fuctos of the groud paraagetc state of Dy os DyAl ca be expressed Jz> otato: Γ > > > > Γ > > > > Γ > > > > Γ > > > > Fg. 3. The result of calculato of eergy level posto vs. teperature of the fe electroc structure of d 9 electroc cofgurato of Dy os DyAl L,S,Lz,Sz> space uder the fluece of tra-atoc sp-orbt couplg, ter-atoc self-cosstet olecular agetc feld ad Crystal Electrc Feld (CEF). The olecular feld splts these states at T<T C. The value of the olecular feld chages ad at T0 (absolute zero) ol 3.T ad ts drecto s alog crystal drecto [100], whch s the x-axs our CEF potetal pcture fro fg.. I ths codto, the wave fucto of the groud sglet gets the for: Γ > > > > The fluece of the exteral agetc feld appled alog dfferet crystal drectos for the structure of the lowest electroc states s show fg.5 ad fg.. The coparso betwee the effects of the exteral agetc feld appled alog dfferet drectos reveals the appearace of a aoaly the level structure f a exteral feld s appled alog crystal drecto [110]. Ths aoaly correspods to the specfc heat curve fro fg.7, E-ISSN: Volue 1, 017

6 coputed the sae exteral feld drecto. The specfc heat of DyAl (spec. heat of f 9 electroc syste +Debye crystal lattce copoet) show fg.7 was calculated for the sae exteral feld codtos as experetal easureets fro ref.[3]. Excellet agreeet betwee calculated specfc heat ad experetal data s also cofred f the exteral agetc feld s appled alog drecto [100]. The results of ths calculato ad referece easureets fro ref.[3] are show fg. 8. level structure calculated for exteral agetc feld ths drecto does ot cota aoales ad s slar to calculatos perfored for ext alog the easy agetzato drecto [100], as show fg. 5. S pecfc heat calculated these codtos s show fg. 8. Fg. 5. Groud eergy level structure vs. teperature calculated for f 9 electroc syste of Dy os DyAl uder fluece exteral agetc feld ext 1T appled alog drecto [100]. Paraeters of CEF used calculatos ad olecular feld factor are show. Fg. 7. Calculated f-electro copoet of olar specfc heat (eq.1) wth Debye crystal lattce copoet (θ350k) vs. teperature for Dy os DyAl, uder the fluece of a exteral agetc feld appled alog drecto [110]. For coparso, the set shows experetal data fro [3] captured the sae exteral agetc feld codtos as sulated. Fg.. Groud eergy level structure vs. teperature calculated for f 9 electroc syste of Dy os DyAl uder fluece of exteral agetc feld ext T appled alog drecto [110]. Paraeters of CEF used calculatos ad olecular feld factor are show. Calculatos of electroc structure uder the fluece of a exteral agetc feld appled alog drecto [111] were also perfored. Eergy Fg. 8. Calculated f-electro copoet of olar specfc heat (eq.1) wth Debye crystal lattce copoet (θ350k) vs. teperature of Dy os DyAl, uder the fluece of a exteral agetc feld appled alog drecto [100]. For coparso, the set shows experetal data fro [3] captured the sae exteral agetc feld codtos as sulated. E-ISSN: Volue 1, 017

7 crystal alged alog the sae drecto, take fro [3]. The sold les are cogruet wth the experetal data calculated for the ext T ad ext 5T. Fg.8. Calculated f-electro copoet of olar specfc heat (eq.1) wth Debye crystal lattce copoet (θ350k) vs. teperature of Dy os DyAl, uder the fluece of a exteral agetc feld ext fro 0 to 10T appled alog drecto [111]. Collected data of specfc heat akes t possble to calculate sotheral etropy chage S(T, ext ) accordg to eq.1, the sae ethodology as used by experetalst [3-7]. Isotheral etropy chage calculated wth varous exteral agetc felds appled alog all a drectos of cubc structure s preseted fgs. 9, 10 ad 11. The referece data take fro experetal easureet [3] s show the sets. Fg. 10. Calculated sotheral etropy chage vs. teperature (eq.1) for varous values of exteral agetc felds fro 0 t o T wth step 0.1T, appled alog drecto [110] of DyAl crystal lattce. Iset: black tragles show sotheral etropy chage obtaed fro experetal data extracted fro the agetzato; epty crcles show specfc heat for a DyAl sgle crystal alged alog the sae drecto, take fro [3]. Fg.9. Calculated sotheral etropy chage of felectroc syste vs. teperature (eq.1) of Dy os DyAl, uder the fluece of varous values of exteral agetc feld fro 0 t o 5T, wth step 0.T appled alog drecto [100]. Iset: sotheral etropy chage obtaed fro experetal data; black tragles -extracted fro the agetzato, crcles - extracted fro specfc heat, for the DyAl sgle Fg.11. Calculated sotheral etropy chage of DyAl vs. teperature (eq.1) for varous values of exteral agetc feld fro 0 t o T wth step 0.1T, appled alog dagoal drecto [110]. The asotropc behavor of calculated theroagetc propertes s reflected the agetocrystalle asotropy costat calculatos. E-ISSN: Volue 1, 017

8 The results of K (T) calculatos accordg to eq.17 the absece of exteral agetc feld are show fg. 1. Fg. 1. Calculated accordg to eq.17, agetocrystalle asotropy costats K 1,K ad K 3 vs. teperature, calculated for Dy os DyAl uder the fluece of CEF ad olecular agetc feld. Fg. 1. Calculated x copoets (sold les), z- copoets (dashed les) of total agetc oet total (doted les) vs. teperature, calculated for Dy os DyAl uder fluece of CEF, olecular agetc feld, ad varous values of exteral agetc feld fro 0 to 7T wth step 0.T appled alog drecto [110]. For copleteess, agetc oet calculatos vs. teperature uder varous exteral agetc feld codtos were perfored. The results of (T, ext ) are preseted fg.13, fg.1, fg.15. Fg. 13 clearly cofrs drecto [100] as a easy agetzato axs. The appled exteral agetc feld alog ths drecto cofrs perfect parallel drectos of agetc vector ad exteral agetc feld (ote: cubc syetry [100][010][001]). Fg. 13. Calculated x,y,z-drectoal copoets of total agetc oet vs. teperature, calculated for Dy os DyAl uder fluece of CEF ad olecular agetc feld ad varous values of exteral agetc feld fro 0 to 7T wth step 0.T appled alog drecto [100]. Fg.15. Calculated x copoets (sold les), z- copoets (dashed les) of total agetc oet total (doted les) vs. teperature, calculated for Dy os DyAl uder fluece of CEF, olecular agetc feld, ad varous values of exteral agetc feld fro 0 to 7T wth step 0.T appled alog drecto [111]. Magetc oet calculated a exteral agetc feld parallel to drecto [110] reveals uusual behavor of the drectoal copoet of the total oet. Slar behavor of agetc oet drectos was reported ref. [3-5] 5 Coclusos We perfored calculatos for DyAl theroagetc propertes wth the ATOMIC MATTERS MFA coputato syste. The local syetry of the Dy os s cubc, whch sgfcatly splfes the aalyses. All the E-ISSN: Volue 1, 017

9 calculatos were perfored wth oly cubc CEF paraeters ad olecular feld factor ol. The excellet agreeet obtaed betwee the calculated theroagetc propertes ad the referece data cofrs the effectveess of our theoretcal approach. Workg wth ATOMIC MATTERS MFA revealed ts hgh usefuless. The vsual for of calculato results, full 3D teractve CEF potetal vsualzato, tutve tools for coveto ad ut recalculato, ad the ablty to copare data results all ake t easy for users to utlze the power of the applcato very effectvely. I cocluso, we cofr that ATOMIC MATTERS MFA s a u que applcato that cobes a package of tools for correctly descrbg the physcal propertes of atoc-lke electro systes subjected to electroagetc teractos real aterals. Ths s a accurate tool for calculatg the propertes of os uder the fluece of the electrostatc potetal of defable syetry ad both exteral ad ter-oc agetc felds take as a ea feld approxato a agetcally ordered state. Ths paper s the frst a seres devoted to the RAl (Rrare earth) copouds faly o-free paraeters calculatos. The obtaed set of paraeters wll be used for calculatos of heavy rare earth eleets: Gd, Tb, Ho, Er, T crystal structure RAl. The olecular feld factor wll be recalculated accordg to de Gees scalg. Refereces: [1] Rafal Mchalsk, Jakub Zygadło. Descrbg the Fe Electroc Structure ad Predctg Propertes of Materals wth ATOMIC MATTERS Coputato Syste. Poceedgs: 18 th It. Cof. o Physcs, Matheatcs ad Coputer Scece - Duba UE [] Rafal Mchalsk, Jakub Zygadło, Theral depedeces of sgle oc, agetc propertes of aterals ordered state, calculated wth ATOMIC MATTERS MFA coputato syste Poceedgs: 7 th IIF-IIR Iteratoal Coferece o Magetc Refrgerato at Roo Teperature, Therag VII - Toro ITALY, 11-1 Septeber 01. [3] A. L. La, A. O. Tsokol, Karl A. Gscheder Jr., Vtalj K. Pecharsky, Thoas A. Lograsso, ad Deborah L.Schlagel, P hys. Rev. 7(005) 003. [] H.G. Purws ad A. Leso, Adv. Phys. 39 (1990) 309. [5] V. K. Pecharsky, K. A. Gscheder, A. O. Pecharsky, ad A. M. Tsh, Phys. Rev., (001) 10. [] P. J. vo Rake, N. A. de Olvera,E. J. R. Plaza,V. S. R. de Sousa,. P. Alho, A. Magus G. Carvalho, S. Gaa ad M. S. Res, Joural Of Appled Physcs 10 (008) [7] L.A. Gl, J.C.P. Capoy, E.J.R. Plaza ad M.V. de Souza, Joural of Magets ad Magetc Materals 09 (01) 5 9. [8] Magetc ad agetoresstve propertes of cubc Laves phase HoAl sgle crystal, M. Patra, S. Majudar ad S. Gr, Y. Xao, T. Chatterj (011) arxv: [codat.trl-sc] [9] A. laut, R. Mchalsk, M. Kocor, Z. Ropka, A. J. ara ad R.J. Radwask, Acta Physca Poloca 3 (003) [10] Software webste: [11] C. Rudowcz, C. Y Chug, The geeralzato of the exteded Steves operators, J. Phys.: Codes. Matter 1 (00) [1] M. T. Hutchgs, Sold State Phys. 1 (New York (19) 7. [13] J.J.M. Frase, R.J. Radwask; Magetc propertes of bary rare-earth Hadbook of Magetc Materals Vol 7. Edted K.H.J.ushow (1993) [1] R. J. Radwańsk, R. Mchalsk, Z. Ropka, A. laut., Physca 319 (00) [15].Szpuart ad P.A.Ldgard, J. Phys. F: Metal Phys.. Vol. 9. No [1] P.G.de Gees J.Phys.Radat 3 (19) [17]. G. Wyboure, Syetry Prcples ad Atoc Spectroscopy, J. Wley ad Sos, New York (1970). [18] A. Abraga ad. leaey, EPR of Trasto Ios, Claredo Press, Oxford (1970). E-ISSN: Volue 1, 017